Parametric Output Error Based Identification and Fault Detection in Structures Under Earthquake Excitation

Transcription

1 Parametric Output Error Based Identification and Fault Detection in Structures Under Earthquake Excitation J.S. Sakellariou and S.D. Fassois Department of Mechanical & Aeronautical Engr. GR 265 Patras, Greece June 2

2 Output Error Identification And Fault Detection In Civil Structures 1 TALK OUTLINE 1. Introduction 2. The Lumped-Parameter Building Model 3. Output Error Based Structural Identification 4. Fault Detection and Identification 5. Concluding Remarks

3 Output Error Identification And Fault Detection In Civil Structures 2 1. Introduction The General Problem Dynamic identification and fault detection and identification of civil structures under earthquake excitation. Problem Significance The structural identification and fault detection & identification under earthquake excitation is important for: dynamic analysis - modal parameter extraction structural health monitoring - fault diagnosis prediction & control Note that earthquake excitation is attractive for large civil structures. Overall aim: The assessment & improvement of the dynamical characteristics and safety of civil structures

4 Output Error Identification And Fault Detection In Civil Structures 3 Problem Characteristics The earthquake excitation and its characteristics introduce a number of technical difficulties due to: its transient nature its limited duration (limited data records) its limited frequency content its non-stationarity (in terms of both amplitude and frequency) & the presence of non-stationary noise its strength may lead the structure into a non-linear operating regime Coping with these characteristics requires caution & and proper selections.

5 Output Error Identification And Fault Detection In Civil Structures 4 Literature Survey A. Structural Identification Classical frequency domain (spectral) methods are not applicable to the non-stationary stochastic case. Evolutionary spectral density type methods (Staszewski, 1998). ARX & ARMAX time domain methods are not applicable. Adaptive methods (time-domain based on the EKF or alternative recursive schemes) are applicable but limited to slowly-varying nonstationarities and suboptimal accuracy (Ghanem and Shinozuka, 1995; Popescu, 1995; Loh and Tou, 1995; Kirkegaard et al., 1996).

6 Output Error Identification And Fault Detection In Civil Structures 5 B. Fault Detection & Identification Methods based on changes in the structure s stiffness matrix using FEM updating and dynamic/static test data (Hajela and Soeiro, 1989; Hearn and Testa, 1991; Capecchi and Vestroni, 1999;). (difficulties: static testing; complete eigenmode information requires extensive instrumentation/testing; a stiffness element may have contributions from several structural elements; stochastic effects often unaccounted for). Methods based on changes in the structure s modal characteristics (Yao, 1988; Salawu, 1997). (difficulties: most require complete modal information; extensive instrumentation/testing; fault magnitude estimation problematic; fault identification may be impossible from pattern changes; stochastic effects often unaccounted for). Methods based on damage indices (Dipasquale and Çakmak 199; Wiliams and Sexsmith, 1995; Garcia et al., 1999). (difficulties: measurement in every storey; extensive instrumentation; ambiguous correlation with the actual fault).

7 Output Error Identification And Fault Detection In Civil Structures 6 Study Objectives The formulation of an effective Output Error (OE) based method for the dynamic identification and fault detection of structures under earthquake excitation. Intended characteristics: Suitable for non-stationary excitation and measurement noise. Achieving high accuracy & resolution with limited data records. Effective fault detection & identification in a stochastic environment. Effectiveness with a single transfer function measurement (two sensors). Capability for fault magnitude estimation.

8 Output Error Identification And Fault Detection In Civil Structures 7 2. The Lumped-Parameter Building Model Figure 1: Schematic diagram of the 6 storey building. m i = kg, k i = KN/m, c i = Ns/m.

9 Output Error Identification And Fault Detection In Civil Structures 8 Theoretical building characteristics (Ẍ6/X transfer func- Table 1: tion). f n (Hz) ζ(%) Residue j j j j j j.824 The building longitudinal dynamics are described by the differential equation: M ẍ + C ẋ + K x = Q x x = [x 1 x 2 x 3 x 4 x 5 x 6 ] T : longitudinal storey displacement vector x = [x ẋ ] T : input vector (x : longitudinal ground displacement) Q : input shaping matrix The above model is reformulated in state space form as follows: ż = A z + B x o y = C z

10 Output Error Identification And Fault Detection In Civil Structures 9 z = [z 1... z 12 ] T : state vector, y = [ ψ 1 ψ 1... ψ 6 ψ ] T 6 A = 1 k 1+k 2 m 1 c 1+c 2 m 1 k 2 m 1 c 2 m 1 1 k 2 m 2 c 2 m 2 k 2+k 3 m 2 c 2+c 3 m 2 k 3 m 2 c 3 m 2 1 k 3 m 3 c 3 m 3 k 3+k 4 m 3 c 3+c 4 m 3 k 4 m 3 c 4 m 3 1 k 4 m 4 c 4 m 4 k 4+k 5 m 4 c 4+c 5 m 4 k 5 m 4 c 5 m 4 1 k 5 m 5 c 5 m 5 k 5+k 6 m 5 c 5+c 6 m 5 k 6 m 5 c 6 m 5 1 k 6 m 6 c 6 m 6 k 6 m 6 c 6 m 6 B = [ 1 ]T C = I(12) The expression for the 6 th floor acceleration is: ẍ 6 [t] = k 1 m 1 ψ 6 + c m 1... ψ 6

11 Output Error Identification And Fault Detection In Civil Structures 1 3. Output Error Based Structural Identification Earthquake Excitation Earthquake recorded in Patras, Greece, in Non-stationary behaviour - Energy concentration in low frequencies ( - 1 Hz) Duration of strong excitation 6.52 sec Maximum acceleration.19g Sampling rate f s = 2Hz Recording time 37.2 sec N = 748 samples Frequency (Hz) (a) Displacement (m) 6 x (b) Time (sec) Figure 2: Seismic displacement (a) and its spectrogram (b) (f s = 25Hz).

12 Output Error Identification And Fault Detection In Civil Structures 11 Modeling of the earthquake displacement Adaptive Recursive AutoRegressive Moving Average (RARMA) modeling based upon the Recursive Maximum Likelihood (RML) algorithm is used along with non-parametrically estimated innovations variance (sliding non-causal window) (Fouskitakis and Fassois, 2). A RARMA(3,8) model with forgetting factor λ =.975 is obtained. RARMA(3,8): x[t] 3 i=1 φ i [t] x[t i] = a[t] 8 i=1 θ i [t] α[t i] x[t] : ground displacement a[t] : innovations (uncorrelated) sequence E{a[t]} = E{a 2 [t]} = σα[t] 2 φ i [t] : time-varying AR parameter θ i [t] : time-varying MA parameter This RARMA(3,8) model is used for the generation of artificial ground displacement signals to be used in the Monte Carlo experiments.

13 Output Error Identification And Fault Detection In Civil Structures 12 Simulation of the Building Dynamics Integration via step invariance discretization at f s = 2Hz Low pass filtering (f c = 1Hz) of the excitation and resulting responses. Signal resampling at f s = 25Hz. The resulting signals are N = 926 samples long. The response (6 th storey acceleration) is corrupted with non-stationary uncorrelated noise at the 5% N/S ratio. Frequency (Hz) (a) Acceleration (m/s 2 ) Time (sec) Figure 3: 6 th floor acelleration (a) and its spectrogram (b) (nominal structure). (b)

14 Output Error Identification And Fault Detection In Civil Structures 13 The Output Error model Output Error (OE) identification aims at the estimation of a discrete-time structural model of the form, OE(na, nb) model : ẍ 6 [t] = B(B) A(B) x [t] + e[t] x [t]: ground displacement ẍ 6 [t]: noise-corrupted 6th storey acceleration e[t]: Output error / non-stationary disturbance B: Backshift operator (B x[t] = x[t 1]) A(B) = 1 + a 1 B a na B n a : AutoRegressive (AR) polynomial B(B) = b + b 1 B b nb B n b : EXogenous (X) polynomial n a : AR order n b : X order The main advantage of the OE model is that it is capable of accounting for noise effects without resorting on an explicit noise representation overcoming problems with non-stationary measurement noise.

15 Output Error Identification And Fault Detection In Civil Structures 14 Parameter Estimation Parameter estimation is based upon minimization of a quadratic functional of the Output Error. Thus: ˆθ N = arg min θ J(θ) = arg min θ 1 N N 1 t= e 2 [t] θ = [ a 1 a 2... a na. b b 1 b 2... b nb ] T Optimization is based upon the Levenberg-Marquardt scheme (Ljung, 87). The estimate is asymptotically Gaussian with mean equal to the true parameter vector and specific covariance: θ o : true parameter vector N(ˆθN θ o ) N (,P) (N) Model Order Selection Model order selection is based on the successive estimation of increasingly higher order models and: Examination of the achieved Output Error cost J(ˆθ) Modal Dispersion Analysis (Lee and Fassois, 1993) provides the energy contribution of each mode in the measured signal unnecessary (overfitted) modes are detected by their small dispersions

16 Output Error Identification And Fault Detection In Civil Structures 15 System Identification Results The identification procedure leads to an OE(6,6) model (b ) with J = Output Error Criterion [6,5] [6,6] [6,7] [6,8] [6,9] [7,5] [7,6] [7,7] [7,8] [7,9] [8,5] [8,5] [8,5] [8,5] Output Error model orders (na,nb) [8,5] [9,5] [9,6] [9,7] [9,8] [9,9] Figure 4: structure). Output Error for some Output error models (nominal

17 Output Error Identification And Fault Detection In Civil Structures 16 Monte Carlo Identification experiments (nominal structure) 15 2 Magnitude (db) Phase (deg) Frequency (Hz) Frequency (Hz) Figure 5: Bode diagrams of the estimated OE(6,6) models ( ) and the theoretical structure (- - -) (Ẍ6/X transfer function; 2 Monte Carlo experiments; nominal structure). Mode Natural Frequencies (Hz) Damping factors (%) Dispersions (%) (2.18 ± ) 1.21 (1.21 ± ) 5.1 (5.91 ±.23) (6.41 ± ) 3.56 (3.56 ± ) (39.12 ± 1.62) (1.26 ± ) 5.7 (4.6 ± ) 6.95 (54.97 ± 1.85) True (Estimated ± Standard Deviation)

18 Output Error Identification And Fault Detection In Civil Structures Fault Detection and Identification Structural Faults The faults considered represent reductions in the elasticity characteristics of each storey. A fault mode is defined as the continuum of faults of all possible magnitudes occurring in a particular storey. Six fault modes are presently defined: F 1 α, F 2 α,..., F 6 α α: fault magnitude (stiffness reduction) Several structural health monitoring Monte Carlo experiments are considered corresponding to: F (no fault) F 3.5 (5% reduction in K 3 ) F 1.5 (5% reduction in K 1 ) F 3.2 (2% reduction in K 3 ) F 1.2 (2% reduction in K 1 ) The effects of the faults on the considered ẍ 6 [t]/x [t] transfer function are minimal.

19 Output Error Identification And Fault Detection In Civil Structures Magnitude (db) 1 2 Phase (deg) Frequency (Hz) Frequency (Hz) Figure 6: Theoretical Bode diagram of the nominal structure ( ), the structure under F 3.5 (- - -), and the structure under F 3.2 (...) (Ẍ6/X transfer function). Acceleration (m/s 2 ) Acceleration (m/s 2 ) Acceleration (m/s 2 ) Time (sec) Figure 7: The 6 th floor acceleration in the nominal (a) and damaged (k 3 reduced by 5% and 2%) (b), (c) cases. (a) (b) (c)

20 Output Error Identification And Fault Detection In Civil Structures 19 Fault Detection Methodology Fault detection is based upon comparison of the interval estimator of the model parameter vector in the nominal and current states of the structure. Let C i represent the confidence interval (at the α =.5 level) of the i-th scalar parameter θ i : C i = (ˆθi 1.96ˆσ θi, ˆθ ) i ˆσ θi The structure is declared as healthy if all confidence intervals of the current parameter vector are not disjoint with those of the nominal parameter vector. Otherwise the structure is declared as damaged. C i C u i = { } for some i otherwise Fault detected No fault detected C i C u i : conf. interval of θ i (nominal state of the structure) : conf. interval of θ i (current state of the structure) { } : empty set C i C i C u i C u i No fault Fault

21 Output Error Identification And Fault Detection In Civil Structures 2 Fault Identification Methodology Once a fault has been detected, fault identification (localization) and magnitude estimation are accomplished via the Geometric Approach (Sadeghi & Fassois, 1998). This consists of seven stages: Stage 1: An initial feature vector θ is selected, e.g. θ = [AR parameters] T Stage 2: The feature vector is transformed into a coordinate system in which information compression may be best achieved: s = U T θ (P = U Λ U T : covariance) Stage 3: A reduced, transformed, feature vector s M is obtained by specifying the allowable information loss. θ = M s j u j + N s j u j j=1 j=m+1 }{{}}{{} reduced error feature vector = [U M.U N M ] s M... s N M Information is expressed in terms of logarithmic entropy: H R (s) = N ν j = Var[s j ]/ N j=1 j=1 ν j log 2 ( ν j ) Var[s j ] (j = 1, 2,..., N)

22 Output Error Identification And Fault Detection In Civil Structures 21 Stage 4: A stochastic feature space, spanned by the 1st & 2nd order moments of the transformed & reduced feature vector s M, is constructed. θ = [ µ T s M, Cov[s M ] ] T (ρ dimensional) Stage 5: Fault mode geometric representations (hyperplanes). A geometric representation of each fault mode Fα i (i = 1, 2,..., 6) within the stochastic feature space is constructed with the aid of experiments performed with a simulation model. The i-th fault mode is thus represented by the hyperplane equation: g i (θ K ) = θ K1 + ω1 i θ K ωρ 1 i θ Kρ ωρ i = (ω1, i ω2, i..., ωρ) i estimated via linear regression. Stage 6: Fault mode identification (localization). The vector θ u K corresponding to the current fault is estimated. Its distances from the fault mode hyperplanes are estimated by optimizing the Lagrangian: L(θ K, γ) = D(θ K, θ u K ) + 2γ gi (θ K ) D(θ K, θ u K ) : squared distance function The current fault θ u K is then identified as belonging to the fault mode with the hyperplane of which the distance is minimal.

23 Output Error Identification And Fault Detection In Civil Structures 22 Stage 7: Fault magnitude estimation. A measure of the fault magnitude is obtained by evaluating the distance between the current fault and the nominal system: α D 1/2 (θk o, θu K ) Figure 8: Principle of the geometric approach (θk u : current fault; g i (θ K ) = : i-th fault mode subspace; θk o : nominal-unfailed system)

24 Output Error Identification And Fault Detection In Civil Structures 23 Fault Detection and Identification Results Two test cases: F 3.5 and F Confidence Intervals a 1 a 2 a 3 a 4 a 5 a 6 AR Parameters Figure 9: Fault detection results for the F 3.5 (...) and F 3.2 (- - -) faults (the confidence intervals of the parameters of the nominal structure are indicated by solid line; single experiments).

25 Output Error Identification And Fault Detection In Civil Structures 24 Initial feature vector: θ = [α 1 α 2... α 6 ] T Transformed & reduced feature vector: s = [s 1 s 2 s 3 ] T θ K = [µ s1 µ s2 µ s3 σ 2 s 1 σ 2 s 2 σ 2 s 3 ] T Entropy percentage Feature Vector Dimension Figure 1: Normalized logarithmic entropy of the transformed feature vectors for every damage level, versus their dimension (F 3 case).

26 Output Error Identification And Fault Detection In Civil Structures 25 injected faults F 3 α α = Parameter θ k Parameter θ k Parameter θ k1 1.8 (a) Distance %.5 5% Magnitude (%) (b) Figure 11: (a) 3-dimensional intersection of the F 3 hyperplane along with projections (+) of various fault points; (b) the distance function versus fault magnitude for the F 3 fault mode (two examples of fault magnitude estimation for the F 3.5 and F 3.2 faults are indicated).

27 Output Error Identification And Fault Detection In Civil Structures 26 9 x 1 3 Fault F (a) Distance minimal distance F 1 F 2 F 3 F 4 F 5 F 6 Distance Fault F 3.2 minimal distance F 1 F 2 F 3 F 4 F 5 F 6 Fault Modes (b) Figure 12: Distances of the current fault point from each hyperplane: (a) Current fault F 3.5; (b) current fault F 3.2 (single experiments).

28 Output Error Identification And Fault Detection In Civil Structures 27 Monte Carlo Experiments (2 realizations each) Structure Correct Detection False Detection No Fault 2 F F F F Fault Correct Identification False Identification F F F F 1.2 2

29 Output Error Identification And Fault Detection In Civil Structures Concluding Remarks A stochastic time-domain Output Error based method for the dynamic identification and fault detection/identification of structures under earthquake excitation was introduced. The method was shown to overcome difficulties associated with: the excitation s transient nature & limited duration (N=926 samples long data records were used) the excitation s limited frequency range (-1 Hz) the excitation s strong non-stationarity & the presence of nonstationary random measurement noise (N/S = 5%) the need for multiple measurement (only the ground and 6 th storey accelerations were measured) The method was shown to achieve: Statistically consistent and highly accurate estimates of the structure s modal parameters. Effective fault detection, identification and magnitude estimation in the presence of random non-stationary noise. Effectiveness with faults as small as 5 % stiffness reduction.